In any data interpolation or extrapolation, we want the extended data to behave like the original data. And, in regions where there is no observed data, the extrapolated data should drop away in a fashion consistent with its spectrum determined from the known region. We will see that a filter like (a-2, a-1, 1, a1, a2) fails to do the job. We need to keep an end value constrained to ``1,'' not the middle value.
In chapter we learned about the interpolation-error filter (IE filter), a filter constrained to be ``+1'' near the middle and consisting of other coefficients chosen to minimize the power out. The basic fact about the IE filter is that the spectrum out tends to the inverse of the spectrum in, so the spectrum of the IE filter tends to the inverse squared of the spectrum in. The IE filter is thus not a good weighting function for a minimization, compared to the prediction-error (PE) filter, whose spectrum is inverse to the input. To confirm these concepts, I prepared synthetic data consisting of a fragment of a damped exponential, and off to one side of it an impulse function. Most of the energy is in the damped exponential. Figure 10 shows that the spectrum and the extended data are about what we would expect. From the extrapolated data, it is impossible to see where the given data ends.
Figure 10 Top is synthetic data with missing data represented by zeros. Middle includes the interpolated values. Bottom is the filter, a prediction-error filter which may look symmetric but is not quite.
For comparison, I prepared Figure 11. It is the same as Figure 10, except that the filter is constrained in the middle. Notice that the extended data does not have the spectrum of the given data--the wavelength is much shorter. The boundary between real data and extended data is not nearly as well hidden as in Figure 10.
Figure 11 Top is synthetic data with missing data represented by zeros. Middle includes the interpolated values. Bottom is the filter, an interpolation-error filter.
Next I will pursue some esoteric aspects of one-dimensional missing-data problems. You might prefer to jump forward to section , where we tackle two-dimensional analysis.